Exotic bases of functions for fourier decomposition?

It is useful to express a function as a combination of sin(kx) and cos(kx) terms because differentiating those functions twice is the same as applying a scale factor (of -k^(2)). In other words, they are eigenfunctions of the operator d^(2)/dx^(2) with the eigenvalue -k^(2).

The whole idea behind Sturm-Liouville theory is that you can change the operator to something else that's amenable to the problem you're trying to solve, which will give you a different set of eigenfunctions that are specific to your problem. Then, you can express other functions as a combination of those eigenfunctions, as a basis.
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You should look at the Laplace transform which is used in control theory. Or wavelet transforms which are used to compress images (although modern standards don't use them much anymore). There is also Hough and Radon transform used in CT scan but I don't know/remember much about them.
In quantum mechanics every problem is associated with a complete set of function on which you decompose a generic wavefunction. For example if you have an harmonic oscillator you can get the most out of decomposing your wavefunction on the basis of Hermite polynomials or if you have a point particle on a sphere you can decompise the radial parts of the wavefunction on the spherical harmonics
In addition to the other answers, it might be interesting for you to also look at Gabor transform and frames. They are not (usually) orthonormal, but also give decompositions of functions.

EDIT: One other thing I haven't seen mentioned is Chebysev-Fourier expansion. It also has some nice properties.
Keep in mind that an important property of the Fourier transform is that it switches (up to some factor of i) between pointwise multiplication and differentiating. I'm not sure if this is the raison d etre of the transform, but it leads to pretty important statements such as the Fourier inversion theorem
You should look into finite elements discretization. It's a way of creating a non orthogonal basis set that captures arbitrary geometries and produces sparse equations.
Can you tell me what you’re even talking about? What can you use this for in the real world?
Sincerely an idiot.
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Wavelets
If you don't make it trigonometric or complex exponential, then it's not really the Fourier transform anymore. Then it's just some other change of basis transformation that is convenient for the problem at hand. For instance, wavelet decompositions are an example of that, and these are used in image compression.

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